Philippe Jacquet ; Charles Knessl ; Wojciech Szpankowski - Counting Markov Types

dmtcs:2768 - Discrete Mathematics & Theoretical Computer Science, January 1, 2010, DMTCS Proceedings vol. AM, 21st International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms (AofA'10) - https://doi.org/10.46298/dmtcs.2768
Counting Markov TypesArticle

Authors: Philippe Jacquet 1; Charles Knessl 2; Wojciech Szpankowski 3

  • 1 High performance communication
  • 2 Department of Mathematics, Statistics and Computer Science [Chicago]
  • 3 Department of Computer Science [Purdue]

The method of types is one of the most popular techniques in information theory and combinatorics. Two sequences of equal length have the same type if they have identical empirical distributions. In this paper, we focus on Markov types, that is, sequences generated by a Markov source (of order one). We note that sequences having the same Markov type share the same so called $\textit{balanced frequency matrix}$ that counts the number of distinct pairs of symbols. We enumerate the number of Markov types for sequences of length $n$ over an alphabet of size $m$. This turns out to coincide with the number of the balanced frequency matrices as well as with the number of special $\textit{linear diophantine equations}$, and also balanced directed multigraphs. For fixed $m$ we prove that the number of Markov types is asymptotically equal to $d(m) \frac{n^{m^{2-m}}}{(m^2-m)!}$, where $d(m)$ is a constant for which we give an integral representation. For $m \to \infty$ we conclude that asymptotically the number of types is equivalent to $\frac{\sqrt{2}m^{3m/2} e^{m^2}}{m^{2m^2} 2^m \pi^{m/2}} n^{m^2-m}$ provided that $m=o(n^{1/4})$ (however, our techniques work for $m=o(\sqrt{n})$). These findings are derived by analytical techniques ranging from multidimensional generating functions to the saddle point method.


Volume: DMTCS Proceedings vol. AM, 21st International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms (AofA'10)
Section: Proceedings
Published on: January 1, 2010
Imported on: January 31, 2017
Keywords: saddle point method,multidimensional generating functions,linear diophantine equations,Markov types,integer matrices,[INFO.INFO-DS] Computer Science [cs]/Data Structures and Algorithms [cs.DS],[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO],[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM],[INFO.INFO-CG] Computer Science [cs]/Computational Geometry [cs.CG]
Funding:
    Source : OpenAIRE Graph
  • Collaborative Research: Information Theory of Data Structures; Funder: National Science Foundation; Code: 0830140
  • Information Transfer in Biological Systems; Funder: National Science Foundation; Code: 0800568
  • Emerging Frontiers of Science of Information; Funder: National Science Foundation; Code: 0939370

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